Four games about quantities that multiply

Log
Space

Each one puts a true reading beside a flattering one, and lets you find out which you had been trusting.

The idea all four are built on

Hold the average still. Widen the spread.

A bet you take over and over, where your money is multiplied by the result each time. The average outcome below never changes, no matter where you put the slider. Only the spread changes — and the rate you actually compound at collapses anyway.

Spread of outcomes

Forty rounds · vertical scale is logarithmic

The four · they open in place

Cross reference · eighteen ideas and where each one is taught

Every idea, and every game that argues it.

The same handful of facts about multiplication turn up in four different costumes. Open any one to see where it appears, what it looks like there, and jump straight to it — several of these drop you into the exact sector or case that makes the point.

Start here if the ideas are new

A Primer on Log Space Everything the four games are about, in order The ruler, then what multiplying does to you, then why one sequence is not a crowd, then how evidence stacks. Five things you can handle rather than only read: a ratio ruler where the geometric mean sits at the visible midpoint, a drag calculator, a curve showing where growth peaks against stake, and a belief rail in decibans. Each part ends with a jump into the game that makes the point. Read

Further afield

Drag in the Complex Plane The same ½σ², arrived at a completely different way What volatility drag becomes when the quantity that multiplies is a complex number. One identity covers both cases — E[e^sX] = exp(sμ + ½s²σ²) — and the whole difference between a convexity bonus and decoherence is that i² = −1. Log space stops being a line and becomes a cylinder; variance starts rotating things as well as scaling them; the mean can fall below the median, which the positive reals never permit; and the phase turns out to have no time average at all. Three live figures, and a table where the simulation fails in exactly the way the mathematics says it must. Read

The wider library · same mathematics, other rooms

Three more, built separately.

These are not games and they do not mount inside this page — they are their own sites, opened in a new tab. The lab in particular goes considerably further than anything here: the four games teach the drag, and it explains where the drag comes from.

Living in Log Space Twenty-nine live models, all on one master knob The most complete thing in this collection, and the argument the four games only gesture at: volatility drag is not a fact about markets, it is a fact about curvature. Part One builds the intuitions from scratch across nine panels — two rulers, the up-10%-then-down-10% asymmetry, the coin-flip ergodicity game, a derivation of the −½σ² tax, and why the walk turns lognormal. Part Two runs one calculation, E[f(X)] − f(E[X]) ≈ ½·f″(μ)·σ², through twenty models across physics, biology, information theory and epidemiology — sunlight through broken cloud, a fading channel, Arrhenius kinetics, stochastic resonance — including the convex cases where the curvature runs the other way and the fluctuation pays you. Open the lab Log Ladder Order-of-magnitude intuition, drilled on a single ruler Fourteen rounds with a lesson after every answer, or sixty seconds with none. Place 248,000 on a log ruler, find the halfway point between 20 and 12,000, slide an actual slide rule, read five octaves up from 261.6 Hz. Scoring is in log units, so being ×2 too high costs exactly what ×2 too low does, and your typical miss is reported as a multiplier rather than a percentage. A linear-view toggle collapses six decades into the left edge, which is the fastest argument for the log ruler there is. Play The Long/Short Desk The same mathematics, as somebody’s job Seven pages of hedge-fund decision-making with every number computed live and every symbol clickable for a worked example: Kelly sizing and the growth curve g(f) = f·μ − ½f²σ², beta hedging, Black–Scholes derived from first principles rather than quoted, arbitrage strategies simulated from their own equations, and portfolio construction. It also carries its own Volatility Drag page, which is the bridge from the games to the desk. Open the desk

The reasoning

Working Notes Every number that had to be settled, and how Thirty-two entries: the mathematics each game rests on, the numbers that could have gone another way and what settled them, the claims that turned out not to be supported, and the defects that ran perfectly while teaching nothing. Written as the games were built, and measured against the shipped code rather than worked out on paper. Read

Four single-file games and one shell, assembled by a build script that patches each game into a mountable module — so each game still has exactly one copy of itself, in its own repository, and this app is generated rather than forked.

Only one game is mounted at a time. Its stylesheet is injected on launch and removed on exit, which is how four games that all define .panel and #app coexist in one document without a single renamed class.

Press Shift + Escape at any time to come back here.

Source: octave · logdrive · deciban · ergodic · logspace · volatility-drag-lab · log-ladder · long-short-desk-html